# Modifications in the Schwarz-Christoffel formula

I am reading Ahlfors's Complex Analysis book, chapter $$6.2.2$$; The Schwarz-Christoffel Formula, and I have some questions in the exercises.

Firstly, the Schwarz-Christoffel formula is given as follows:

Theorem. The functions $$z=F(w)$$ which map the unit disk $$|w|<1$$ conformally onto polygons with (interior) angles $$\alpha_k \pi$$ ($$k=1,...,n; 0<\alpha_k<2$$) are of the form $$F(w)=C \int _0 ^w (w-w_1)^{-\beta_1}...(w-w_n)^{-\beta_n} dw +C'$$, where $$\beta_k=1-\alpha_k$$, the $$w_k$$ are points on the unit circle, and $$C,C'$$ are constants.

$$(1)$$ Show that the $$\beta_k$$ may be allowed to be $$-1$$. What is the geometric interpretation?

$$(2)$$ If the vertex of the polygon is allowed to be at $$\infty$$, what modification does the formula undergo? If in this context $$\beta_k=1$$, what is the polygon like?

In $$(1)$$, if the $$\beta_k=-1$$, then what do I have to justify? Also, if $$\beta_k=1$$ then $$\alpha_k=2$$ so the polygon would have a slit; is this the desired geometric interpretation?

Next, in $$(2)$$, I don't have any idea of modification. If $$\beta_k=1$$ in this case, then I see that the polygon would be a half-plane. Am I right?

I am having a hard time with these. Any help will be appreciated. Thanks!

You're correct about the first question. The formula still works for $$\alpha_k = 2 \pi$$ and gives two fully or partially overlapping segments as part of the boundary.
If $$\alpha_k \leq 0$$, then $$w_k$$ is mapped to $$\infty$$ since $$(w - w_k)^{\alpha_k/\pi - 1}$$ is a non-integrable singularity. If the angle between two lines at $$\infty$$ is defined as minus the angle at their finite intersection point, the formula also works for polygons with a vertex (or several vertices) at $$\infty$$.
Since we have to distinguish between inner and outer angles, the angle at $$\infty$$ for parallel rays (a $$\Pi$$-shaped boundary) is either $$0$$ or $$-2 \pi$$. The angle at $$\infty$$ for antiparallel rays (a $$Z$$-shaped boundary) is $$-\pi$$.
All this is exactly the same for the mapping from the upper half-plane to a polygon, but in that case we can eliminate one of the factors by choosing $$w_k = \infty$$.